Topology Optimization Problem Research Articles

High-order complex Fourier numerical manifold method for improving the optimization of cracked structures

In this paper, a combination of the high-order numerical manifold method with material interpolation is established with the goal of improving the optimization of cracked structures. Complex Fourier shape functions, known for their inherent advantages, are utilized as weight functions in the numerical manifold method. By implementing high-order analysis, the occurrence of checkerboard patterns is effectively avoided. In addition, a modified sensitivity filtering technique is introduced to address the issue of mesh dependency and minimum length scale in the final topology. This technique is based on the notion of a manifold element. Furthermore, a methodology is introduced to represent void regions within the design domain without the need for passive elements. This methodology involves describing an algorithm to accurately determine the manifold element and the corresponding element area within the final topology. Numerical problems of topology optimization are presented to examine all the aforementioned advantages. The findings indicate that the proposed algorithm for topology optimization does not exhibit any instances of checkerboard patterns. Moreover, the final topology of the curved void regions does not show any zigzag patterns. The above method demonstrates a commendable rate of convergence, and the final topology of the cracked regions can be conveniently simulated.

Enhancement layout optimisation of grid structures with stability constraints

Grid structures have found widespread use in civil and mechanical engineering owing to their ability of covering long spans with an outstanding strength-to-weight ratio. However, their shallow structural depth and slender members make them susceptible to buckling failures. Therefore, it is crucial to enhance the stability of grid structures. An under-explored way to achieve this is by optimally configuring enhancement materials. This paper addresses the challenges of identifying the optimal enhancement layouts for grid structures by solving a density-based topology optimisation problem. The problem formulation is based on the SIMP approach, minimising compliance while satisfying constraints on material volume and structural stability. The stability constraint is expressed using buckling load factors obtained from a linear eigenvalue analysis. Two types of enhancement schemes, involving in-plane and out-of-plane elements, are developed for the single-layer base grid structures. The efficiency of the proposed method is evaluated by performing enhancement layout optimisation for a flat and three curved square grid structures, subject to a uniformly distributed vertical load and a half-span load. The optimisation results shed light on the impact of in-plane and out-of-plane enhancements on the stability of grid structures, providing insights on optimally configuring enhancement materials to achieve desired stability. In addition, comparative analysis between the case studies indicates that for curved structures, a higher curvature normally leads to greater achievable stability. In contrast, the flat grid structure exhibits the broadest range of customisable stability due to its primarily bending-dominant load-carrying mechanism. Furthermore, the outcomes of geometric non-linear analysis confirm the load-carrying capacity of the optimal structures, with non-linear critical loads typically showing a positive correlation with linear buckling loads.

A Q-learning driven competitive surrogate assisted evolutionary optimizer with multiple oriented mutation operators for expensive problems

Surrogate-assisted evolutionary algorithms (SAEAs) prevail in the solution of computationally expensive optimization problems. However, with the growth of problem scale and complexity, the high-dimensional problem space greatly restricts their effectiveness and applicability. This paper proposes a Q-learning driven competitive surrogate-assisted evolutionary optimizer with multiple oriented mutation operators (CSEO-MOMO) to improve the prediction effectiveness and accuracy of SAEAs in different scenarios of optimization problems. CSEO-MOMO constructs isomorphic models with different complexities during iteration. The Q-learning method is employed to pick the proper model for prediction based on the state feedback of the iterative population. To explore the search space more comprehensively and increase the probability of finding a better solution, two competitive sub-populations coevolve with personalized mutation operators, of which different oriented optimal tractive base vectors are nominated to guide the mutation direction. In addition, a tailored infilling strategy concerning both the performance of the iterative individuals and their spatial distribution is used to winnow out the individuals with great potential for real evaluation. A suit of high-dimensional CEC benchmarks of different properties and an openly shared topology optimization problem are used for the effectiveness and efficiency verification of CSEO-MOMO. The experimental results reveal that CSEO-MOMO trumps seven excellent SAEAs in solving high-dimensional complex problems and possesses better accuracy and robustness under various fitness landscape scenarios. (The relevant MATLAB code of CSEO-MOMO is publicly available in the first author’s GitHub repository: https://github.com/qinna-zhu/CSEO-MOMO.)

Data-driven and physics-based methods to optimize structures against delamination

Composite materials and multi-material components often fail at their internal interfaces/adhesive joints, and hence special attention should be given to such catastrophic delamination events to guarantee the system’s functional requirements. So far, however, the majority of structural topology optimization problems have focused on optimal distribution of the bulk materials by considering interfaces as perfectly bonded. This motivates the introduction of optimization methods that explicitly take into account the role of the material interfaces to optimize structures against delamination. In this work, we propose a data-driven heuristic optimization approach for the identification of optimal cohesive interfaces with linearly graded fracture properties to increase the ability of the composite structure to withstand peeling. Moreover, for given cohesive interface properties, we investigate the applicability of the physics-based Solid Isotropic Material with Penalty (SIMP) topology optimization approach to optimize the internal structure of a substrate in problems where the stress field is affected by interface delamination.

Buckling mode constraints for topology optimization using eigenvector aggregates

Buckling-constrained structural design problems have conventionally prioritized optimizing the buckling load factor with less consideration given to the buckling mode shape. In this work, mode shape constraints are imposed within a topology optimization problem using an eigenvector aggregate constraint that is a weighted sum of homogeneous quadratic functions of the linearized buckling eigenvectors. A generalized formulation of the eigenvector aggregate is introduced, extending previous work. A new adjoint-based derivative evaluation technique is derived that is valid even in the presence of repeated eigenvalues. Numerical examples, including a clamped beam, a compressed column, and a square plate, demonstrate the effectiveness of the proposed approach. The results show the ability of the eigenvector aggregate to handle repeated eigenvalues, enable design space exploration, and capture mode shape switching.

Fiber Bundle Topology Optimization for Surface Flows

This paper presents a topology optimization approach for the surface flows on variable design domains. Via this approach, the matching between the pattern of a surface flow and the 2-manifold used to define the pattern can be optimized, where the 2-manifold is implicitly defined on another fixed 2-manifold named as the base manifold. The fiber bundle topology optimization approach is developed based on the description of the topological structure of the surface flow by using the differential geometry concept of the fiber bundle. The material distribution method is used to achieve the evolution of the pattern of the surface flow. The evolution of the implicit 2-manifold is realized via a homeomorphous map. The design variable of the pattern of the surface flow and that of the implicit 2-manifold are regularized by two sequentially implemented surface-PDE filters. The two surface-PDE filters are coupled, because they are defined on the implicit 2-manifold and base manifold, respectively. The surface Navier-Stokes equations, defined on the implicit 2-manifold, are used to describe the surface flow. The fiber bundle topology optimization problem is analyzed using the continuous adjoint method implemented on the first-order Sobolev space. Several numerical examples have been provided to demonstrate this approach, where the combination of the viscous dissipation and pressure drop is used as the design objective.

Multi-objective topology optimization method for multi-axis random vibration based on hybrid cellular automata

Considering the lack of effective solutions for multi-axis random vibration topology optimization problems, and recognizing that multi-axis random vibration excitation is the most common loading conditions experienced by structures during their operational life, this paper proposes a multi-objective topology optimization method for multi-axis random vibration. By combining Hybrid Cellular Automata (HCA) and Analytic Hierarchy Process (AHP), this method aims to enhance the vibration resistance of structures and extend their fatigue life during the topology optimization phase. To validate the effectiveness and engineering practicality of this method, two engineering cases were designed: a cantilever beam case and an automotive steering knuckle case. Multi-objective topology optimization were conducted with objectives including mass, stiffness, and vibration intensity metric (root-mean-square stress). The two cases demonstrate that the control group's topology model exhibits weaker resistance to random vibrations and shorter fatigue life. In contrast, the validation group's topology model introduces an additional load path near the excitation point, effectively dissipating the impact of vibration excitation. This enhances the structure's vibration resistance and significantly extends its fatigue life.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

Topological optimization in 3D-magnetostatics: development of adjoint methods using the equations of magnetic moments

Purpose The purpose of this paper is mainly to develop the adjoint method within the method of magnetic moment (MMM) and thus, to provide an efficient new way to solve topology optimization problems in magnetostatic to design 3D-magnetic circuits. Design/methodology/approach First, the MMM is recalled and the optimization design problem is reformulated as a partial derivative equation-constrained optimization problem where the constraint is the Maxwell equation in magnetostatic. From the Karush–Khun–Tucker optimality conditions, a new problem is derived which depends on a Lagrangian parameter. This problem is called the adjoint problem and the Lagrangian parameter is called the adjoint parameter. Thus, solving the direct and the adjoint problems, the values of the objective function as well as its gradient can be efficiently obtained. To obtain a topology optimization code, a semi isotropic material with penalization (SIMP) relaxed-penalization approach associated with an optimization based on gradient descent steps has been developed and used. Findings In this paper, the authors provide theoretical results which make it possible to compute the gradient via the continuous adjoint of the MMMs. A code was developed and it was validated by comparing it with a finite difference method. Thus, a topology optimization code associating this adjoint based gradient computations and SIMP penalization technique was developed and its efficiency was shown by solving a 3D design problem in magnetostatic. Research limitations/implications This research is limited to the design of systems in magnetostatic using the linearity of the materials. The simple examples, the authors provided, are just done to validate our theoretical results and some extensions of our topology optimization code have to be done to solve more interesting design cases. Originality/value The problem of design is a 3D magnetic circuit. The 2D optimization problems are well known and several methods of resolution have been introduced, but rare are the problems using the adjoint method in 3D. Moreover, the association with the MMMs has never been treated yet. The authors show in this paper that this association could provide gains in CPU time.

Simplification of Electrode Profiles for Piezoelectric Modal Sensors by Controlling Gap-Phase Length

This paper presents a method to optimize and simplify the electrode profile of a piezoelectric modal sensor. At the same time that the electrode profile is optimized to maximize the charge, a null-polarity phase is introduced. This gap-phase is modeled using the normalized norm of the spatial gradient of the polarity profile along with a two-step filtering and projection technique. High-order vibration modes generate a complex polarization profile that makes the manufacturing process difficult. The novelty of the proposed method is the addition of a constraint on the length of the interface in the topology optimization problem. This constraint simplifies the optimal designs and facilitates the fabrication process. Several examples show the simplified electrode profile that maximizes the electrical charge produced by a vibration mode, while reducing the number of different polarization regions by means of the gap-length constraint.

DEMFFA: a multi-strategy modified Fennec Fox algorithm with mixed improved differential evolutionary variation strategies

The Fennec Fox algorithm (FFA) is a new meta-heuristic algorithm that is primarily inspired by the Fennec fox's ability to dig and escape from wild predators. Compared with other classical algorithms, FFA shows strong competitiveness. The “No free lunch” theorem shows that an algorithm has different effects in the face of different problems, such as: when solving high-dimensional or more complex applications, there are challenges such as easily falling into local optimal and slow convergence speed. To solve this problem with FFA, in this paper, an improved Fenna fox algorithm DEMFFA is proposed by adding sin chaotic mapping, formula factor adjustment, Cauchy operator mutation, and differential evolution mutation strategies. Firstly, a sin chaotic mapping strategy is added in the initialization stage to make the population distribution more uniform, thus speeding up the algorithm convergence speed. Secondly, in order to expedite the convergence speed of the algorithm, adjustments are made to the factors of the formula whose position is updated in the first stage, resulting in faster convergence. Finally, in order to prevent the algorithm from getting into the local optimal too early and expand the search space of the population, the Cauchy operator mutation strategy and differential evolution mutation strategy are added after the first and second stages of the original algorithm update. In order to verify the performance of the proposed DEMFFA, qualitative analysis is carried out on different test sets, and the proposed algorithm is tested with the original FFA, other classical algorithms, improved algorithms, and newly proposed algorithms on three different test sets. And we also carried out a qualitative analysis of the CEC2020. In addition, DEMFFA is applied to 10 practical engineering design problems and a complex 24-bar truss topology optimization problem, and the results show that the DEMFFA algorithm has the potential to solve complex problems.

Multi-material topology optimization with a direct meshless local Petrov–Galerkin method applied to a two-dimensional boundary value problem

ABSTRACT This work presents a novel approach for the solution of multi-material topology optimization problems of elastic structures coupling the Direct Meshless Local Petrov–Galerkin (DMLPG) method with the Gradual Bi-direction Evolutionary Structural Optimization (gBESO) method. In recent years, meshless methods have been used in several engineering fields, showing some advantages compared to mesh-based methods. DMLPG is a truly meshless method, which achieves results with good accuracy and computational efficiency, avoiding the numerical integration of complicated shape functions by adopting low-degree polynomials. In the proposed algorithm, DMLPG is used to obtain smooth nodal displacements, strains and stresses, and gBESO updates the structural geometry based on design sensitivity values of the compliance while it gradually increases the Young's modulus of the material. The optimization problem minimizes the compliance objective function subject to volume constraints. Numerical examples demonstrate the applicability and validity of the technique.

A novel decoupled approach combining invertible cross-entropy method with Gaussian process modeling for reliability-based design and topology optimization

Design optimization considering the presence of uncertainties in parameters poses an extremely challenging problem. The source of difficulties comes with reliability-based formulations, where addressing the probabilistic problem exhausts the large computing efforts for failure estimations of the structure violating limit-state functions (LSFs). This paper proposes a novel decoupled approach for effectively solving reliability-based design optimization (RBDO) problems, namely an invertible cross-entropy (iCE) method advantageously combined with a Gaussian process regression (GPR) model, termed as iCE-GPR. The GPR model is applied to approximate the spectrum of LSFs under random parameters. Furthermore, to enhance the accurate prediction of the system failure probability, an active learning process is applied to systematically refine the GPR model by adding new learning points in the region with the largest uncertainty and high-reliability sensitivity through the maximization of an expected feasibility function (EFF). Based on the updated GPR model, the failure probability is estimated by a cost-effective cross-entropy (CE) method without any calls to the actual performance function. To perform the decoupling optimization process with the reliability analysis, the novel iCE, based on the CE method, is developed to update the most probable point (MPP) assigned for the next deterministic optimization process in determining the new optimal design. The method iteratively performs the deterministic optimization process based on the MPP underpinning LSFs sequentially updated by the active learning process. The proposed iCE-GPR method fast-converges the optimal design and significantly alleviates computational burdens associated with reliability analyses. The proposed method is also applied to solve a reliability-based topology optimization (RBTO) problem. Four numerical examples for both the RBDO and RBTO problems are provided to illustrate efficiency and robustness of the proposed iCE-GPR method.

Topology optimization based on improved DoubleU-Net using four boundary condition datasets

Topology optimization is extensively employed in engineering design. However, applying deep learning to solve topology optimization problems faces challenges owing to limited training data and model adaptability to various boundary conditions. To address these challenges, this research has employed a solid isotropic material with a penalization method to produce a dataset of 400,000 2D samples covering four boundary conditions. Each boundary condition includes two resolution datasets. Moreover, an improved DoubleU-Net model is proposed for real-time topology optimization, ensuring high-accuracy predictions. On the test dataset, the average Intersection over Union (IoU) accuracies of the proposed model for the four boundary conditions above are 93.26%, 96.71%, 96.35% and 97.38%. This research also investigated the impact of different structural datasets on the model's generalization capability. The experimental results indicate that the model trained on random structures exhibits excellent testing performance and demonstrates robust generalization capability.

Mapping structural topology optimization problems to quantum annealing

Mapping structural topology optimization problems to quantum annealing

Topology optimization of broadband acoustic transition section: a comparison between deterministic and stochastic approaches

This paper focuses on the topology optimization of a broadband acoustic transition section that connects two cylindrical waveguides with different radii. The primary objective is to design a transition section that maximizes the transmission of a planar acoustic wave while ensuring that the transmitted wave exhibits a planar shape. Helmholtz equation is used to model linear wave propagation in the device. We utilize the finite element method to solve the state equation on a structured mesh of square elements. Subsequently, a material distribution topology optimization problem is formulated to optimize the distribution of sound-hard material in the transition section. We employ two different gradient-based approaches to solve the optimization problem: namely, a deterministic approach using the method of moving asymptotes (MMA), and a stochastic approach utilizing both stochastic gradient (SG) and continuous stochastic gradient (CSG) methods. A comparative analysis is provided among these methodologies concerning the design feasibility and the transmission performance of the optimized designs, and the computational efficiency. The outcomes highlight the effectiveness of stochastic techniques in achieving enhanced broadband acoustic performance with reduced computational demands and improved design practicality. The insights from this investigation demonstrate the potential of stochastic approaches in acoustic applications, especially when broadband acoustic performance is desired.

A new density filter for pipes for fluid topology optimization

This study presents a new density filter for a pipe-shaped structure and its application to fluid topology optimization. A simple and straight pipe-shaped structure for fluid is preferred for many engineering purposes rather than the perplex manifold structure provided by the topology optimization method. To determine an optimal pipe structure for fluid, we develop a new density filter and apply it to fluid topology optimization. Hence, the original spatially varying design variables of the fluid topology optimization are modified based on the pipe density filter. Subsequently, the filter design variables, including a uniform pipe wall thickness and adjusted cross-section, are used for artificial pseudo-rigid bodies in fluid topology optimization. An additional constraint is imposed to maintain a nearly uniform pipe thickness. Several numerical examples are solved to demonstrate the validity of the present pipe density filter for fluid topology optimization problems minimizing the energy dissipation of the fluid and controlling the particles suspended in the fluid.

Robust topology optimization for transient dynamic response minimization

This paper presents a novel framework for topology optimization (TO) of structures subjected to uncertain transient loading. Random transient uncertain but bounded loading is modeled in the form of an ellipsoid convex model. A transformation matrix is defined with uncertainty in input parameters based on orientation and lengths of semi-major and semi-minor axes of the ellipsoid. Latin hypercube sampling is utilized to generate samples inside the normalized space. These samples are then converted to a physical space using a transformation matrix. The generalized-α method is employed to compute the dynamic response of the system subjected to time-dependent loading. A density-based TO formulation is used to minimize dynamic response. The robust topology optimization problem is defined as bi-criteria optimization to minimize both the mean and variance across all deterministic cases within the ellipsoidal bounds. The solid isotropic material with penalization method is employed for the interpolation of material stiffness. Helmholtz filter is used to stabilize the problem numerically. The optimality criteria method is used to update the design variables. The proposed scheme’s effectiveness and the efficiency of the optimized designs are checked using a set of numerical examples. For most of the cases, robust design exhibits a substantial reduction of approximately 70%–75% in standard deviation when compared with deterministic design. The results demonstrate the proposed framework’s ability to reduce structure transient response under uncertain but bounded regions.

Robust topology optimization of multi-material structures based on the material-field series-expansion method

The performance of composite structures can be dramatically improved through the synergy between different constituent materials. However, the topology optimization of multi-material structures poses a heavy computational burden, which becomes more significant when load uncertainty is considered. This article proposes an efficient method to address the problem of robust topology optimization of multi-material structures. Employing the property that the material-field series-expansion method can significantly reduce the number of design variables, the proposed method improves the computational efficiency of the robust optimization problem. Meanwhile, the modification of the original multi-material interpolation model improves the quality of the optimized designs. Monte Carlo simulation is used to treat the load uncertainty, but it is decoupled from the optimization process based on linear elasticity theory to reduce computation cost. The effectiveness of the proposed method is demonstrated by numerical examples, which show that robust design results can be efficiently generated by this method.

Multi-scale concurrent topology optimization of cellular structures with multiple microstructures for minimizing dynamic response in the time domain

This paper proposes a concurrent topology optimization method for optimizing the structures that are periodically filled with multiple microstructures excited by dynamic response in the time domain. At the macroscale, different microstructures are considered as different materials. To generate the various distribution of different microstructures, a multi-material interpolation approach based on the Solid Isotropic Material with Penalization (SIMP) is integrated. At the microscale, the energy-based homogenization method (EBHM) is employed to determine the macroscopic effective properties of the microstructures. All macroscale elements with identical material are represented by a distinct microstructure. For the proportional damping model, the HHT-α is implemented as a time integration technique to obtain the dynamic response of multi-scale and assumed multi-material structures. The objective function of the topology optimization problem is to minimize the mean dynamic compliance. Combined differentiate-then-discretize method with the adjoint variable method, the sensitivity analysis applies the gradient-based Zhang-Paulino-Ramos Jr. (ZPR) algorithm or Method of Moving Asymptotes (MMA) to update the design variables under multiple constraints in the time and space-discretized system. The proposed approach is numerically performed through 2D and 3D examples to demonstrate its effectiveness.